Multicomponent density-functional theory for electrons and nuclei

Transcription

1 Multicomponent density-functional theory for electrons and nuclei Thomas Kreibich Institut für Theoretische Physik, Universität Würzburg, Am Hubland, D Würzburg, Germany Robert van Leeuwen Department of Physics, Nanoscience Center, University of Jyväskylä, FI Survontie 9, Jyväskylä, Finland E. K. U. Gross Institut für Theoretische Physik, Freie Universität Berlin, Arnimallee 14, D Berlin, Germany Received 26 September 2006; revised manuscript received 18 October 2007; published 1 August 2008 We present a general multicomponent density-functional theory in which electrons and nuclei are treated completely quantum mechanically, without the use of a Born-Oppenheimer approximation. The two fundamental quantities in terms of which our theory is formulated are the nuclear N-body density and the electron density expressed in coordinates referring to the nuclear framework. For these two densities, coupled Kohn- Sham equations are derived, and the electron-nuclear correlation functional is analyzed in detail. The formalism is tested on the hydrogen molecule H 2 and its positive ion H 2 +, using several approximations for the electron-nuclear correlation functional. DOI: /PhysRevA PACS number s : E, A, w I. INTRODUCTION Density-functional theory DFT is among the most successful approaches to calculate the electronic structure of atoms, molecules, and solids. In its original form 1,2, DFT always invokes the Born-Oppenheimer approximation: One is supposed to calculate the electron density r which is in one-to-one correspondence with the static potential of fixed nuclei. In a recent Letter 3 we introduced a multicomponent density-functional theory MCDFT for the complete quantum treatment of many-particle systems consisting of electrons and nuclei. With this theory it is possible to describe from first principles physical phenomena that depend on a strong coupling between electronic and nuclear motion. MCDFT thereby extends the widely applied densityfunctional formalism for purely electronic properties, opening up a new field of applications, such as the first-principles calculation of electron-phonon coupling in solids 4, which is a key ingredient in the description of superconductivity 5 8 and polaronic motion 9,10. The quantum treatment of the nuclear motion in molecules or solids is essential in situations that from a Born-Oppenheimer BO viewpoint must be described by a superposition of different BO structures. This is, for instance, the case in floppy molecules 11, orin so-called switchable molecules 12, which are in a superposition of an open and a closed state after a laser excitation. Apart from treating such various phenomena, the MCDFT presented here also paves the way for time-dependent extensions of the theory 13 15, which would enable one to calculate the coupled electronic and nuclear dynamics of manyparticle systems, within linear response and beyond. Indeed, some preliminary steps toward the description of the coupled ionization and dissociation dynamics of molecules in strong laser fields have already been taken 13,16,17. The purpose of the present work is twofold. First, we want to give an extended and detailed description of the theory that was briefly described in our Letter. Second, we want to investigate in detail some approximate density functionals for the electron-nuclear correlation and see how they perform. To do this the formalism is tested on the hydrogen molecule and its positive ion. The paper is organized as follows. In Sec. II we first introduce the basic formalism and discuss the Hohenberg-Kohn theorem and the Kohn-Sham equations in a multicomponent theory in which the electron density is defined with respect to a coordinate frame attached to the nuclear framework and in which the diagonal of the nuclear density matrix appears as a new variable. In Sec. III we perform an analysis of the several energy functionals and of the resulting potentials in the Kohn-Sham equations. Furthermore, the connections between the effective potential of the nuclear Kohn-Sham equation and the Born-Oppenheimer energy surface is analyzed. In Sec. IV we apply our formalism and test several approximate forms for the electronnuclear correlation functional for the case of the hydrogen molecule and its positive ion. Finally, in Sec. V we present our conclusions. II. BASIC FORMALISM A. Discussion of the Hamiltonian We consider a system composed of N e electrons with coordinates r j r= and N n nuclei with masses M 1,...,M Nn, charges Z 1,...,Z Nn, and coordinates denoted by R R=. By convention, the subscripts e and n refer to electrons and nuclei, respectively, and atomic units are employed throughout this work. In nonrelativistic quantum mechanics, the system is described by the Hamiltonian where Ĥ = Tˆ n R= + Ŵ nn R= + Û ext,n R= + Tˆ e r= + Ŵ ee r= + Û ext,e r= + Ŵ en R=,r=, 1 N n Tˆ n = =1 2 2M, /2008/78 2 / The American Physical Society

2 KREIBICH, VAN LEEUWEN, AND GROSS N e Tˆ e = j=1 2 j 3 2 denote the kinetic-energy operators of the nuclei and electrons, respectively, and N n Ŵ nn = 1 2, =1 Ŵ ee = 1 2 N e Ŵ en = j=1 i,j=1 i j Z Z R R, N e 1 N n =1 r i r j, Z r j R represent the interparticle Coulomb interactions. We emphasize that no BO approximation has been assumed in 1 ; the Hamiltonian of Eq. 1 provides a quantum mechanical description of all, i.e., electronic and nuclear, degrees of freedom. In contrast to the standard approach using the BO approximation, the interactions between electrons and nuclei are therefore treated within Ŵ en, Eq. 6, and do not contribute to the external potentials. Truly external potentials representing, e.g., a voltage applied to the system, are contained in N n Û ext,n = U ext,n R, 7 =1 N e Û ext,e = u ext,e r j. 8 j=1 Defining electronic and nuclear single-particle densities conjugated to the true external potentials 7 and 8, a MCDFT formalism can readily be formulated on the basis of the above Hamiltonian 18. However, as discussed in 3, such a MCDFT is not useful in practice because the single-particle densities necessarily reflect the symmetry of the true external potentials and are therefore not characteristic of the internal properties of the system. In particular, for all isolated systems where the external potentials 7 and 8 vanish, these densities, as a consequence of the translational invariance of the respective Hamiltonian, are constant. A suitable MCDFT is obtained by defining the densities with respect to internal coordinates of the system 3. To this end, new electronic coordinates are introduced according to where r j = R,, r j R c.m.n j =1,...,N e, 9 R c.m.n ª 1 N n M nuc M R. =1 10 denotes the center of mass c.m. of the nuclei, the total nuclear mass is given by N n M nuc = M, =1 11 and R is the three-dimensional orthogonal matrix representing the Euler rotations 19. The Euler angles,, are functions of the nuclear coordinates R= and specify the orientation of the body-fixed coordinate frame. They can be determined in various ways. One way to define them is by requiring the inertial tensor of the nuclei to be diagonal in the body-fixed frame. The conditions that the off-diagonal elements of the inertia tensor are zero in terms of the rotated coordinates R R R c.m.n then give three determining equations for the three Euler angles in terms of the nuclear coordinates R=. This way of choosing the Euler angles is commonly used within the field of nuclear physics 20 22, but is, of course, not unique. A common alternative way to determine the orientation of the body-fixed system is provided by the so-called Eckart conditions 23 29, which are suitable to describe small vibrations in molecules and phonons in solids 4. A general and very elegant discussion of the various ways the body-fixed frame can be chosen is given in Ref. 30. In this work we will not make a specific choice as our derivations are independent of such a choice. The most important point is that, by virtue of Eq. 9, the electronic coordinates are defined with respect to a coordinate frame that is attached to the nuclear framework and rotates as the nuclear framework rotates. In fact, this transformation comprises two transformations: A first one transforming the space-fixed inertial coordinates into c.m.-fixed relative coordinates, and a second one transforming the c.m.-fixed relative coordinates into body-fixed internal coordinates. The nuclear coordinates themselves are not transformed any further at this point, i.e., R = R, =1,...,N n. 12 Of course, introducing internal nuclear coordinates is also desirable. However, the choice of such coordinates depends strongly on the specific system to be described: If nearequilibrium situations in systems with well-defined geometries are considered, normal or for a solid phonon coordinates are most appropriate, whereas fragmentation processes of molecules are better described in terms of Jacobi coordinates 31. Therefore, keeping a high degree of flexibility, the nuclear coordinates are left unchanged for the time being and are transformed to internal coordinates only prior to actual applications in the final equations that we will derive. Another reason for not introducing any internal nuclear coordinates at this point is to retain simple forms of the equations. In a transformation to internal nuclear coordinates, typically the nuclear center-of-mass and the Euler angles are taken as new variables as well as 3N n 6 internal or shape coordinates Q i These internal coordinates, however, do not have a simple relation to the original N n nuclear coordinates and will therefore lead to a complicated form of the Hamiltonian in the new coordinates. We will therefore delay the use of such transformations until we have derived the final equations

3 MULTICOMPONENT DENSITY-FUNCTIONAL THEORY FOR As a result of the coordinate changes of Eq. 9, the Hamiltonian 1 transforms into Ĥ = Tˆ n R= + Ŵ nn R= + Û ext,n R= + Tˆ e r= + Ŵ ee r= + Tˆ MPC R=,r= + Ŵ en R=,r= + Û ext,e R=,r=. 13 Since we have transformed to a noninertial coordinate frame, mass-polarization and Coriolis MPC terms N n Tˆ MPC ª 1 N e 2 rj R + =1 2M rj j=1 R Tˆ n R= 14 appear. Obviously, Tˆ MPC is not symmetric in the electronic and nuclear coordinates. However, this was not expected since only the electrons refer to a noninertial coordinate frame, whereas the nuclei are still defined with respect to the inertial frame. Therefore, all MPC terms arise solely from the electronic coordinates, representing fictitious forces due to the electronic motion in noninertial systems for a detailed form of these terms within the current coordinate transformation, see 4. The kinetic-energy operators Tˆ e and Tˆ n, the electron-electron and nuclear-nuclear interactions, as well as the true external potential Û ext,n acting on the nuclei are formally unchanged in Eq. 13 and therefore given by Eqs. 2 and 4 with the new coordinates replacing the old ones, whereas the electron-nuclear interaction now reads N e Ŵ en R=,r= = j=1 The quantity N e = j=1 N n =1 N n =1 Z R,, 1 r j R + R c.m.n Z r j R,, R R c.m.n. 15 R = R,, R R c.m.n 16 that appears in Eq. 15 is a so-called shape coordinate 4,30, i.e., it is invariant under rotations and translations of the nuclear framework, R OR= + a = R R=, 17 where O is an arbitrary rotation matrix and a an arbitrary translation vector. The invariance property described in Eq. 17 is simply a consequence of the fact that the Euler angles are defined by giving the vectors R certain values, independent of where the nuclear center of mass was situated in the laboratory frame or how the nuclear framework was orientated. This is, of course, precisely the purpose of introducing a body-fixed frame. For this reason the potential in Eq. 15 that the electrons in the body-fixed frame experience from the nuclei is invariant under rotations or translations of the nuclear framework. As a further result of the coordinate transformation 9, the true external potential acting on the electrons now depends not only on the electronic coordinates, but also on all the nuclear coordinates: N e Û ext,e R=,r= = u ext,e R 1 r j + R c.m.n. 18 j=1 In the chosen coordinate system the electron-nuclear interaction 15 and the external potential 18 remain one-body operators with respect to the electronic degrees of freedom but represent complicated N n -body interactions with respect to the nuclei. We finally discuss some general aspects of our coordinate transformation. If we consider the symmetry properties of our original Hamiltonian of Eq. 1 in the absence of external potentials, we see that it is invariant under simultaneous translations and rotations of all particles, i.e., of both electrons and nuclei. This is no longer true for our transformed Hamiltonian. Since we transformed the electronic coordinates to a body-fixed frame, we find that in the absence of external potentials the transformed Hamiltonian of Eq. 13 is invariant under translations and rotations of nuclear coordinates only. The corresponding ground-state wave function, if it is nondegenerate, will have the same invariance. Let us next consider the permutational symmetry. The ground-state wave function of the original Hamiltonian of Eq. 1 is antisymmetric under the interchange of electronic space-spin coordinates and symmetric or antisymmetric under interchange of nuclear space-spin coordinates of nuclei of the same type, depending on whether they are bosons or fermions. The ground-state wave function of the transformed Hamiltonian of Eq. 13 will also be antisymmetric with respect to the interchange of electronic space-spin coordinates. However, the symmetry properties with respect to the interchange of the nuclear space-spin coordinates depend on the conditions that we choose to determine the Euler angles. If we choose a determining constraint for the Euler angles that is symmetric in the interchange of particles of the same type, then the transformed wave function will retain the permutational symmetry properties of the original wave function. This is, for instance, the case if we determine the Euler angles by the requirement that the nuclear inertia tensor be diagonal. However, if we choose a nonsymmetric constraint, such as the Eckart conditions, then the transformed wave function will have more complicated transformation properties under the interchange of nuclear spin-space coordinates since the interchange of two nuclear coordinates will then also change the Euler angles a detailed account of this topic is given in Ref. 29. This can lead to practical complications but will not affect our general formalism. We finally note that the coordinate transformation we presented here did not aim at a separation of the constants of motion of the system even for the case of isolated systems. In contrast, the transformation 9 was chosen such that the new electronic coordinates reflect the internal symmetry of the system. We thus arrive at a Hamiltonian that naturally lends itself as a starting point for the formulation of a MCDFT, as will be shown in the subsequent sections. B. Definition of the densities As a first step toward the formulation of a densityfunctional theory, one has to define the densities which will

4 KREIBICH, VAN LEEUWEN, AND GROSS serve as the fundamental variables of the theory. Although this seems to be rather straightforward and is normally not discussed at length, a careful definition of the densities is of crucial importance in the current context. As already mentioned above, it is not useful to define electronic and nuclear single-particle densities in terms of the inertial coordinates r and R, since such densities necessarily reflect the symmetry of the corresponding true external potentials, e.g., Galilean symmetry for vanishing external potentials. Therefore, such single-particle densities are not characteristic for the internal properties of the system under consideration. We proceed with the definition of a suitable set of densities, which should satisfy the following requirements: 1 They should be characteristic for the internal properties of the system; in particular, they should be meaningful in the limit of vanishing external potentials. 2 The basic electronic variable should be a single-particle quantity. 3 The treatment of the nuclear degrees of freedom should allow for appropriate descriptions of situations as different as nearequilibrium properties of solids and fragmentation processes of molecules. A set of densities that meets these requirements is given by R= = s, d N er R= s=,r= = 2, r = N e s, d N nr d N e 1 r R= s=,r= = 2, where R= s=,r= = corresponds to the ground state of Hamiltonian 13 and where s= and = denote the nuclear and electronic spin coordinates. These densities are defined with respect to the transformed coordinates R=,r=. In particular, the electronic single-particle density r refers to the bodyfixed molecular frame. In terms of these coordinates, the quantity 20 represents a conditional density, which is characteristic for the internal properties of the system. It is proportional to the probability density of finding an electron at position r as measured from the nuclear center of mass, given a certain orientation of the nuclear framework. Therefore the electronic density calculated through 20 reflects the internal symmetries of the system, e.g., the cylindrical symmetry of a diatomic molecule, instead of the Galilean symmetry of the underlying space. The nuclear degrees of freedom, on the other hand, are described using the diagonal of the nuclear density matrix, Eq. 19. In the absence of external potentials this quantity will have the transformation property OR= + a = R=, 21 where O is a rotation and a a translation vector. Its permutational properties will depend on the choice of the bodyfixed frame as discussed in the previous section. The quantity R= allows us to set up a general as well as flexible formalism, which will be applicable to a large variety of situations. In an actual application, one may at a later stage further contract this quantity to obtain reduced density matrices or, depending on the physical situation, introduce more suitable internal nuclear coordinates, which could not be done if single-particle quantities had already been introduced at this point. C. The Hohenberg-Kohn theorem for multicomponent systems In this section, we discuss the extension of the Hohenberg-Kohn theorem to multicomponent systems. In contrast to prior formulations of the MCDFT 18,32 35, this analysis will employ the densities 19 and 20 as fundamental variables. Correspondingly, the starting point of the following analysis is the Hamiltonian 13. In order to formulate a Hohenberg-Kohn- HK- type statement, the Hamiltonian 13 is generalized to where Ĥ = Tˆ + Ŵ + Û + Vˆ, Tˆ = Tˆ n R= + Tˆ e r= + Tˆ MPC R=,r= denotes the total kinetic-energy operator and Ŵ = Ŵ ee r= + Ŵ en R=,r= contains the electron-electron and the electron-nuclear interaction. Furthermore, auxiliary external potentials conjugated to the densities 19 and 20, Vˆ = Vˆ n R= + Vˆ e r=, 25 have been added to the Hamiltonian. We note that, in the transformed coordinates, Vˆ n actually acts as an N n -body operator with respect to the nuclear coordinates, Vˆ n = V n R=, 26 and particularly contains the internuclear repulsion Ŵ nn R=, while Vˆ e is a one-body operator with respect to the bodyfixed electronic coordinates: N e Vˆ e = v e r. j 27 j=1 The true external potentials, on the other hand, are subsumed in Û = Û ext,n R= + Û ext,e R=,r=. 28 Note that the nuclear potential Û ext,n has the same structure as Vˆ n, while the electronic potential Û ext,e acts similarly to the electron-nuclear interaction in the transformed coordinate system. The Hamiltonian 22 and the above defined densities 19 and 20 now provide a suitable basis for the formulation of the multicomponent Hohenberg-Kohn MCHK theorem. It can be summarized by the following statements. 1 Uniqueness. The set of ground-state densities, uniquely determines the ground-state wave function

5 MULTICOMPONENT DENSITY-FUNCTIONAL THEORY FOR =, as well as the potentials Vˆ n=vˆ n,,vˆ e =Vˆ e,. As a consequence, any observable of the static many-body system is a functional of the set of ground-state densities,. 2 MCHK variational principle. The total-energy functional E, ª, Ĥ, 29 is equal to the exact ground-state energy E 0 if the exact densities 0 and 0 corresponding to fixed external potentials Vˆ n,0 and Vˆ e,0 are inserted into the functional. For all other densities, the inequality E 0 E, 30 holds true. This MCHK theorem can be proven by using both the reductio ad absurdum and the constrained search approach, familiar from standard DFT 36. In the following, a generalization of the latter to multicomponent systems will be presented. We start out by defining the functional F, ª min Tˆ + Ŵ + Û,, 31 i.e., we search for the minimum of Tˆ +Ŵ+Û using all properly normalized and symmetrized wave functions yielding a given set of densities,. It must be noted that all the wave functions that we use in the constrained search procedure are now also required to have the correct symmetry properties with respect to interchange of nuclear spacespin coordinates of nuclei of the same type. As we discussed before these symmetry properties depend on the way we define the body-fixed frame. For instance, if we define the body-fixed frame by a diagonalization of the nuclear inertia tensor then the constrained search must be carried out over all wave functions that are antisymmetric in the electronic spin-space coordinates and symmetric or antisymmetric with respect to the interchange of nuclear spin-space coordinates, depending on whether the nuclei are bosons or fermions. If we denote the minimizing state assuming it exists 1 by min,, we realize that F, = min, Tˆ + Ŵ + Û min, 32 is by construction a functional of the densities. We note that, in contrast to the usual DFT, the functional F is not universal since it still depends on the external potentials Û which, as a result of our coordinate transformation, are functions of both R= and r=, as was discussed in connection with Eq In standard electronic DFT, one can prove that the minimum of F exists 59. Using Eq. 32, the total-energy functional is given by E, = F, + d N nr R= V n R= + dr r v e r. 33 The variational principle 30 can now be proven by employing the Rayleigh-Ritz variational principle: E 0 = min Ĥ. 34 Following the constrained search procedure 37 of ordinary DFT, the minimum in 34 is split into two consecutive steps: E 0 = min min Ĥ,, = min, F, + d N nr R= V n R= + dr r v e r = min E,, 35, where the external potentials V n and v e are held fixed during the minimization. For notational simplicity, the primes indicating the transformed coordinates are dropped from now on. By convention, all electronic coordinates are understood to refer to the body-fixed frame. In the second step, we have exploited the fact that all wave functions that lead to the same densities also yield the same external energy. By virtue of the Rayleigh-Ritz variational principle, the minimizing densities are the ground-state densities 0 and 0. Furthermore, any other set of densities will lead to an energy above the true ground-state energy if inserted in the total-energy functional 33. This completes the proof of statement 2. In order to prove the first statement, we reformulate the variational principle 35 according to F, + d N nr R= V n R= + dr r v e r =0. 36 Since the variations can be done independently, Eq. 36 is equivalent to F, R= + V n R= =0, 37 F, + v e r =0. 38 r If the exact densities 0, 0 are inserted, the Euler equations 37 and 38 are satisfied for the true external potentials. If, on the other hand, an arbitrary set of densities, is inserted, Eqs. 37 and 38 define assuming the functional derivatives exist a set of potentials, which reproduce, as ground-state densities. Therefore, the set of densities, uniquely determines the external potentials V n,v e and thus the ground-state wave function = min,

6 KREIBICH, VAN LEEUWEN, AND GROSS Before concluding, a number of remarks are added. 1 As usual, the potentials are uniquely determined up to an arbitrary additive constant, and nondegeneracy of the ground state has been assumed. 2 Similar to purely electronic DFT, the functional F, is defined via Eq. 32 for all, -representable densities, i.e., for all densities obtained according to Eqs. 19 and 20 from a many-body wave function with the right permutational symmetries. The potentials V n,v e are defined for all densities, for which the functional derivatives in Eqs. 37 and 38 exist, i.e., for all interacting V n,v e -representable densities. 3 If vanishing external potentials 18 are considered, the analysis reduces to the one given in 3. D. The Kohn-Sham scheme for multicomponent systems As usual, the HK theorem does not depend on the specific form of the particle-particle interaction. In particular, it can be applied to an auxiliary system which is characterized by Ŵ ee =Ŵ en =Tˆ MPC=0, i.e., the system consists of noninteracting electrons and of nuclei that interact only among themselves. The key assumption in establishing the multicomponent Kohn-sham MCKS scheme is that local effective potentials Vˆ S,n,Vˆ S,e exist such that the ground-state densities of the auxiliary system reproduce the exact ground-state densities 0, 0 of the fully interacting system. If that assumption holds true, the exact ground-state densities are given by 0 R= = R= s= 2, 39 s N e 0 r = j r 2, 40 j=1 where and j are solutions of an N n -particle nuclear and a single-particle electronic Schrödinger equation, respectively: 2 2M + V S,n R= n R= s= =0, v S,e r e,j j r =0. 42 By virtue of the MCHK theorem applied to the auxiliary system, the effective potentials V S,n R= and v S,e r are uniquely determined by the ground-state densities 0, 0, once their existence is assumed. They are given by V S,n R= = V n R= + E U,Hxc,, 43 R= 0, 0 v S,e r = v e r + E U,Hxc, r 0, 0, 44 where for the case of isolated molecules we have V n R= =W nn R= and v e r =0. In this procedure we require the nuclear wave function to have the same symmetry properties under the interchange of nuclei of the same type as the exact wave function of the interacting system this will also be required for the adiabatic connection to be discussed later in the paper. The last terms on the right-hand sides of Eqs. 43 and 44 represent the potentials due to all nontrivial interactions of the system, i.e., they contain the Hartree exchange-correlation Hxc effects of the electron-electron and electron-nuclear interactions as well as mass-polarization and Coriolis effects and the influence of the true external potentials Û. As seen in Eqs. 43 and 44, these potentials are given as functional derivatives of the U, Hxc energy functional defined by E U,Hxc, ª F, T S,n T S,e. 45 This quantity represents the central quantity of the MCDFT and contains all many-body effects except the purely nuclear correlations. We note that, in the case of vanishing external potentials Û 0, the nuclear effective potentials V S,n R= and the conjugated density, i.e., the nuclear density matrix R=, are invariant under translations. Therefore, the nuclear center of mass can be separated off in Eq. 41, reducing the number of degrees of freedom by 3. We will illustrate this procedure in our applications later. In order to derive the above representations of the effective potentials, we consider the energy functional of the auxiliary system introduced above: E S, = T S,n + T S,e + d N nr R= V S,n R= + dr r v S,e r. 46 As noted before, the nuclear-nuclear interaction Ŵ nn is included in the external potential V S,n R=. The noninteracting kinetic-energy functional T S,e is the one familiar from purely electronic DFT, T S,e = min Tˆ e, 47 where the minimization is over all electronic Slater determinants yielding. Similarly, the nuclear kinetic-energy functional is given by T S,n = min Tˆ n. 48 In contrast to the electronic wave function, the nuclear wave function is not a Slater determinant, but a correlated many-body wave function, since it minimizes Tˆ n under the constraint of generating the diagonal of the nuclear N n -particle density matrix. We note that, although is an interacting many-body wave function, T S,n is not the interacting nuclear kinetic-energy functional T n, =, Tˆ n,, since, minimizes Tˆ +Ŵ +Û for given densities,, therefore including all electron-nuclei interactions as well as mass-polarization and Coriolis couplings. Assuming the densities, to be non

7 MULTICOMPONENT DENSITY-FUNCTIONAL THEORY FOR interacting V n,v e -representable, the minimizing states of 47 and 48, i.e., the states minimizing the kinetic energy for given,, are obtained from Eqs. 41 and 42 with the potentials uniquely determined by the Euler equations following from 46 : T S,n + V S,n R= =0, 49 R= T S,e + v S,e r =0. 50 r Returning to the interacting problem, we decompose the functional F, according to Eq. 45. Employing this definition in the variational equations 37 and 38 of the interacting problem and comparing them to the Euler equations 49 and 50, we find that the effective potentials which reproduce the exact densities from the auxiliary system are indeed given by Eqs. 43 and 44. Equations constitute the MCKS system. Since the effective potentials depend on both densities, the MCKS equations 41 and 42 are coupled, reflecting the mutual influence of electrons and nuclei on each other, and have to be solved self-consistently. We emphasize that Eq. 42, although similar to the usual electronic KS equation, does not parametrically depend on the nuclear configuration. Instead, the information on the nuclear distribution is already included through the functional dependence on. Considering the nuclear MCKS equation 41, we again realize its similarity with the conventional nuclear BO equation. Yet no BO approximation has been used to derive Eq. 41. In contrast, since the MCKS scheme provides the exact ground state, all non-bo effects are, in principle, included. Whether or not the non-bo effects are reproduced in practical applications depends, of course, on the quality of the approximations employed for E U,Hxc,. We also note that in the absence of external potentials the potential V S,n R= has the same symmetry properties as the BO-energy surface under rotations and translations, i.e., V S,n OR= + a = V S,n R=. 51 The way it will transform under interchange of like nuclei will depend on the way we choose the body-fixed frame. It is also important to realize that when solving the nuclear equation 41 we must look for the solution R= s= with the lowest energy under the constraint that it has the correct symmetry under interchange of nuclear space-spin coordinates, i.e., the symmetry that was imposed by the constrained search. Like the nuclear BO equation, the nuclear equation 41 is still a many-body equation. Therefore, its solution will, in general, be rather complicated and further simplifications are highly desirable. Typically, one first splits off the nuclear center-ofmass motion and the global rotations of the molecule. Then the remaining nuclear degrees of freedom are transformed to normal coordinates, in terms of which the problem is treated in a harmonic approximation, possibly including anharmonic effects in a mean-field fashion 38. However, due to the generality of the method, different treatments appropriate for different physical situations can be used. III. ANALYSIS OF THE FUNCTIONALS A. Decomposition of the energy functional In the last section, the foundations of the MCDFT were developed. We derived a formally exact scheme, which provides a way to calculate ground-state properties of multicomponent systems. For any practical application, the functional E U,Hxc needs to be approximated. In order to gain more insight into the construction of such an approximation, this section discusses a number of rigorous properties of this functional. Following 34, we start out by decomposing the U,Hxc energy functional 45 in parts associated with its various interactions. To this end, we define the following quantities: F e ª min Tˆ e + Ŵ ee, F en, ª min Tˆ n + Tˆ e + Ŵ,, T MPC, ª min Tˆ n + Tˆ e + Tˆ MPC + Ŵ F en,,, 54 U Hxc, ª F, min Tˆ + Ŵ., 55 The first term represents the electronic functional which, by construction, is identical to the functional F LL of standard electronic DFT, first introduced in 37. Usually, this quantity is split according to where E H e F e = T S,e + E e H + E e xc, is the electronic Hartree functional 56 E e H ª 2 1 dr dr r r, 57 r r e and where the electronic exchange-correlation functional E xc is defined by Eq. 56. In contrast to F e, the second functional F en also includes the nuclear kinetic energy as well as the electron-nuclear interaction, but still neglects masspolarization and Coriolis effects and the influence of the external potential Û. As discussed later on, the functional F en thus includes in particular the effects arising from the electron-nuclear correlation. The first term on the right-hand side of Eq. 54 additionally contains the mass-polarization and Coriolis terms of the kinetic-energy operator. Therefore, the difference between min Tˆ +Ŵ and F en is responsible for mass-polarization and Coriolis effects and thus denoted by T MPC. Similarly, the last term denoted by U Hxc takes care of all effects introduced by the true external potentials Û. Consequently, if no true external fields are applied to the system, U Hxc vanishes identically

8 KREIBICH, VAN LEEUWEN, AND GROSS Inserting Eqs into Eq. 45 leads to where E U,Hxc, = E e H + E e xc + E en Hc, + T MPC, + U Hxc,, 58 E en Hc, ª F en, T S,n F e. 59 Equation 58 provides a decomposition of the Hxc energy functional into its natural contributions. The first part, given e by E H and E e xc, describes the Coulomb interactions among the electrons. It is important to note that these functionals are, by construction, identical to the ones familiar from standard electronic DFT. The electron-electron interaction can therefore be treated in the familiar way, namely, by using the widely investigated and highly successful approximations for the electronic xc energy functional E e xc. The last term of Eq. 58 was constructed to incorporate all effects arising from the presence of true external potentials Û. As already mentioned above, these terms are not of a single-particle form in the transformed coordinate system and have to be treated similarly to the interaction terms. The functional U Hxc, provides a means of dealing with these effects. Similarly, the fourth term of Eq. 58 incorporates all effects due to the mass-polarization and Coriolis terms. At least for ground-state properties, this term is expected to be unimportant and can be neglected in most situations. If such effects are, on the other hand, important in a given physical situation, they can, in principle, be included in the calculation by taking the functional T MPC, explicitly into account. Finally, the Hartree-correlation Hc term E en Hc, contains all effects due to the electron-nuclear interaction. Its analysis will be continued in the next section. The decomposition 58 of the energy functional E U,Hxc is obviously not unique. However, the charm of the above prescription lies in the fact that, first, parts like the purely electronic functionals are already well known such that one can rely on existing approximations for these functional. Second, the functionals contain, by their very construction, just the effect of one specifically chosen interaction. This, in e particular, guarantees that the functionals E Hxc, E en Hc,, and T MPC, are universal in the sense that they do not depend on the external potentials and can therefore be employed for all systems independent of the applied external fields. All effects arising from the external potentials are subsumed in U Hxc,. en B. The electron-nuclear energy functional E Hc Using the decomposition 58, the well-studied electronic Hxc energy functional as well as the at least for groundstate properties presumably negligible mass-polarization and Coriolis contribution were separated off in the functional E U,Hxc. In this section, we discuss the functional E en Hc, which contains the many-body effects due to the electronnuclear interaction. We will derive an equation for this functional that is of a suitable form to be used in our approximations later. To do this we will use the familiar coupling constant integration technique of standard density-functional theory. To begin with, we consider the Hamiltonian Ĥ = Tˆ n + Tˆ e + Ŵ ee + Ŵ en + Vˆ n, + Vˆ e,, 60 where a non-negative coupling constant, scaling the electron-nuclear interaction, has been introduced. As usual, the potential Vˆ =Vˆ n, +Vˆ e, is chosen such that the densities remain fixed: = and =, independent of the coupling constant. Employing the coupling-constant integration technique adapted to the electron-nuclear interaction 34, the electron-nuclear Hc energy functional 59 is rewritten as E en Hc, = min, = 0 = 0 min, 1 1 Tˆ n + Tˆ e + Ŵ ee + Ŵ en =1 Tˆ n + Tˆ e + Ŵ ee + Ŵ en =0 d min,, Tˆ n + Tˆ e + Ŵ ee + Ŵ en min,, d min,, Ŵ en min,,, 61 where min,, denotes the minimizing state of Tˆ n+tˆ e+ŵ ee + Ŵ en generating the given densities,, and a Hellmann-Feynman-type theorem was used in the last step. en Therefore, the electron-nuclear energy functional E Hc is given by E en Hc, = d N nr R= dr W en R=,r min, r R=, where Z W en R=,r = R 1 r R + R c.m.n Z = r R R R c.m.n. The electronic conditional density is defined by r R= ª N e,s d N e 1 r r= =,R= s= 2 / R=, and represents the coupling-constant average of. The conditional density satisfies important sum rules that we will use later for the construction of approximate functionals: N e = dr r R= R=, 65 r = dr= R= r R=. 66 By virtue of Eq. 62, the Hc energy can be interpreted as the electrostatic interaction energy of the coupling-constantaveraged electronic density for a fixed nuclear configuration with the point charges of the corresponding nuclei, averaged over the nuclear distribution. This interpretation will play an

9 MULTICOMPONENT DENSITY-FUNCTIONAL THEORY FOR important role in our later development of approximate functionals. In order to gain further insight into the electron-nuclear Hc energy functional, we now establish a connection between the MCDFT scheme and the conventional BO method which provides a highly successful treatment of electronnuclear correlation. To that end, we decompose the total wave function into an adiabatic product according to min, r= =,R= s= = R= s=, r= = R= s=, 67 where is the nuclear wave function generating the nuclear density matrix and, is an electronic state normalized to one for every nuclear configuration R= s=: d N er, r= = R= s= 2 =1. 68 We note that the decomposition 67 is actually an exact representation of the correlated electron-nuclear wave function and that the factors and are unique 48 up to within an R= s=-dependent phase factor. However, it is important to note that the electronic state is not identical to the usual electronic BO state. Even if non-bo effects were neglected, would not be identical to the electronic BO state since and are required to reproduce a given set of densities, the two electronic wave functions only become equivalent if, is evaluated at the BO densities, = BO, BO. Instead,, is expanded according to, r= = R= s= = a k, R= s= BO R=,k r= =, 69 k where BO R=,k denotes a complete set of BO eigenfunctions corresponding to the electronic clamped-nuclei Hamiltonian Ĥ e ªTˆ e+ŵ ee +Ŵ en. Employing Eq. 68 together with 68 and assuming that, is real, the electron-nuclear Hc energy functional is given from Eq. 59 by E en Hc, = min, Tˆ n + Ĥ e min, T S,n F e = d N nr R= s= 2, Tˆ n s + Ĥ e, e F e, 70 where the index e at the angular bracket indicates that the integration is over electronic coordinates only. Using Eq. 69 we then obtain E en Hc, = s where d N nr R= s= 2 a k, R= s= 2 BO k R= k,l k,l + a k, R= s= BO k Tˆ n BO l e a l, R= s= F e, 71 BO k R= = BO k Ĥ e BO k e 72 represents the kth BO potential-energy surface PES and the index e at the angular bracket indicates that the integration is over electronic coordinates only. On the basis of Eq. 71, one can interpret E en Hc as the potential energy of the nuclei, where the nuclear distribution lies in a potential hypersurface which is composed of adiabatic BO PESs weighted with the coefficients a k, as well as nonadiabatic corrections to it. Of course, the coefficients a k and their functional dependence on the set of densities, are unknown at this point. However, Eq. 71 helps us in gaining a better understanding of the electron-nuclear Hc energy functional and establishes an at least formal link to the BO scheme, which is further exploited when the effective potentials are discussed later on. C. Concerning the true external potentials Similar to the techniques employed in the last section, the coupling-constant integration can be employed to derive an expression for the functional U Hxc, which subsumes the many-body effects arising from the true external field. In analogy to Eq. 62, one obtains U Hxc, = d N nr R= dr U ext R=,r min, r R=, 73 where min again denotes the coupling-constant average with respect to coupling constant of the conditional density min,, corresponding to the states min,, minimizing Tˆ +Ŵ+ Û. We further defined U ext R=,r to be U ext R=,r ª 1 N e U ext,n R= + u ext,e R 1 r + R c.m.n. 74 It has to be noted that the conditional densities appearing in Eqs. 62 and 73 are not identical since the corresponding states min,, and min,, minimize different expressions. This is a direct consequence of the definitions chosen in Eqs. e In particular, this choice guarantees that E Hxc, T MPC, and E en Hc are independent of the true external potential U ext, i.e., these functionals are universal; all effects stemming from U ext are contained exclusively in the functional U Hxc. By virtue of the above discussion, the influence of the true external potential has to be treated similarly to an interaction. As already mentioned above, this complication is an immediate consequence of the necessity to transform to an internal reference system for the formulation of the MCDFT scheme. Of course, in the numerous cases discussing the properties of isolated systems, Û vanishes and the MCDFT formalism reduces to the one given in 3. If, on the other hand, a true external potential is applied to the system, approximations for U Hxc are needed. In the simplest case, the electronic conditional density is replaced by the electronic density, leading to a Hartree-type approximation for U Hxc. Such an approximation will be especially valid in the case of welllocalized nuclei, as discussed later on. D. Analysis of the effective potentials In the last sections, the Hxc energy functional of Eq. 45 was discussed. According to Eqs. 43 and 44, this quantity

10 KREIBICH, VAN LEEUWEN, AND GROSS gives rise to the many-body contributions of the effective MCKS potentials. Explicitly, the U,Hxc potentials are given by V U,Hxc, R= = E U,Hxc,, 75 R= v U,Hxc, r = E U,Hxc,. 76 r Employing Eq. 58, the potentials can also be decomposed into the parts associated with the different interactions, yielding V U,Hxc, R= = V U Hxc, R= + V en Hc, R= + V MPC, R=, 77 v U,Hxc, r = v U Hxc, r + v e H r + v e xc r + v en Hc, r + v MPC, r, 78 where the various potential terms on the right-hand sides of the above equations are defined in analogy to 75 and 76 : The first terms on the right-hand side of Eqs. 75 and 76 represent the influence of the true external potential Û and correspond to the derivatives of U Hxc in Eq. 58. Since the electron-electron interaction is treated employing the wellknown Hxc energy functional E Hxc from standard elec- e tronic DFT, the corresponding potentials v e e H and v xc are also identical to the familiar electronic Hartree and xc potentials. Furthermore, as for the energy functional, the potentials arising from the mass-polarization and Coriolis effects are not expected to contribute significantly at least for ground-state properties. In the following we will concentrate on the Hxc potentials arising from the electron-nuclear energy functional E en Hc. To start with, we consider the nuclear Hc potential, defined by V en Hc, R= = E Hc en,. 79 R= en Employing the representation of E Hc in terms of the coupling-constant-averaged conditional density, Eq. 62, the nuclear potential can be split into two parts, where en, R= = V en cond, R= + V en c,rsp, R= 80 V Hc V en cond, R= ª dr W en R=,r min, r R= 81 is the electrostatic potential due to the electronic conditional density and V en c,rsp, R= ª d N nr R= dr W en R=,r min, r R= R= 82 defines a response-type contribution to the electron-nuclear correlation potential. We note that the conditional potential en completely determines the electron-nuclear Hc energy: V cond E en Hc, = d N nr R= V en cond, R=. 83 In the following analysis we restrict ourselves to situations where the full electron-nuclear wave function min, can be factorized into a nuclear spin function times a remainder not depending on s=. This is exactly true, e.g., for diatomic molecules or when the nuclei are spin-zero bosons. In many other cases, this factorization represents a good approximation. Under these circumstances, the wave function in Eq. 67 can be chosen to be independent of s=, and likewise the expansion coefficients a k in Eq. 69, so that Eq. 71 reduces to en, = d N nr R= E Hc k,l a k, R= 2 k BO R= k,l + a k, R= BO k Tˆ n BO l e a l, R= F e. 84 Comparing Eqs. 83 and 84, the conditional potential 81 can be expressed in terms of the BO PES: V en cond, R= = a k, R= 2 BO k R= + a k, R= k k,l k BO Tˆ n l BO e a l, R= F e. 85 This equation provides a useful tool to interpret the effective nuclear MCKS potential: The first term in 85 is a weighted sum over different adiabatic BO PESs, whereas the second one describes adiabatic and nonadiabatic corrections to it. The last term in Eq. 85, F e, just yields a constant shift and is included in the potential to maintain the same zeroenergy level within the BO and MCKS schemes. Considering the case that the BO approximation accurately describes a specific system, we realize that, in the first sum, only the lowest coefficient a 0 survives and the second sum is negligible, provided the potential is evaluated at the ground-state densities. Therefore, V en Hc R= BO 0 R=, and the nuclear MCKS equation reduces to the nuclear BO equation in the limit considered here. We emphasize, however, that the way to evaluate this potential differs in the MCKS and BO methods. Whereas, in the latter, an electronic equation has to be solved for each nuclear configuration, the MCKS potential is determined by the functional derivative E en Hc, /. Inserting the ground-state densities then yields a potential which, as a function of R=, is very close to the BO potential in the case discussed here. Furthermore, we can conclude that the response part of the nuclear potential, Eq. 82, has

11 MULTICOMPONENT DENSITY-FUNCTIONAL THEORY FOR negligible influence for such systems. If, on the other hand, nonadiabatic effects e.g., close to level crossings are encountered, the coefficients a k in Eq. 85 will, as a function of the nuclear configuration, achieve a natural diabatization. One should also note that the electronic wave function is, in general, complex at points of degeneracy. Therefore, one obtains another contribution to the nuclear potential, which is responsible for Berry-phase effects In addition, the response part of the nuclear potential might contribute appreciably. In summary, Eq. 85 shows that the exact nuclear effective potential reduces to the lowest-energy BO PES, if nonadiabatic contributions can be neglected, but also contains in principle all non-bo effects. Whether or not they can be recovered in an actual application crucially depends, of course, on the level of sophistication of the approximation used for E en Hc. Employing again Eq. 62, the electronic potential due to the electron-nuclear interaction, defined by is given by v en Hc, r ª E Hc en,, 86 r v en Hc, r = d N nr R= dr W en R=,r min, r R=. r 87 This expression appears rather complicated to evaluate. If, however, the nuclear probability distribution is sharply peaked around an equilibrium geometry R= eq, only configurations around R= eq will substantially contribute to the above integral. Then the calculation of the electronic Hxc potential simplifies to E. The limit of classical (pointlike) nuclei In this section, we investigate the limit of classical, i.e., perfectly localized nuclei. Assuming identical zero spin nuclei for the ease of notation, the nuclear density matrix reads class R 1,...,R Nn = 1 R P R,0, 89 N n! P where the sum is over all N n! permutations of the nuclear coordinates and R= 0 denotes the positions where the nuclei are located. Note that by using this classical form of the density matrix we have broken the translational and rotational symmetry of the density matrix as presented in Eq. 21. In the following we investigate the consequences of this form for the diagonal of the nuclear density matrix. First, we consider the electronic density. In terms of the couplingconstant-dependent conditional density, it is given by r = d N nr R= r R=. 90 We recall that r does not depend on the coupling constant, since the external potentials are chosen such that the densities remain unchanged. Inserting Eq. 89 into 90 yields r = r R= 0, 91 i.e., in the limit of classical nuclei, the electronic density is identical to the conditional density evaluated at the positions of the classical nuclei. This quantity, in fact, serves as the basic variable of standard electronic DFT employing the BO approximation: DFT,BO r = r R= 0. We therefore conclude that the MCDFT presented here reduces to the standard formulation of DFT in the limit of classical nuclei. Inserting Eq. 89 and 91 into Eqs. 62 and 73, we readily obtain the expressions for E en Hc and U Hxc in the classical limit: E en Hc class, = dr r W en R= 0,r, 92 v en Hc, r d N nr R= W en R=,r. 88 U Hxc class, = dr r U ext R= 0,r. 93 This potential represents the electrostatic Hartree potential due to the nuclear charge distribution acting on the electrons. Since the nuclear ground-state densities of many molecules are indeed strongly localized functions in other words, the nuclei behave almost classically we expect the Hartree approximation for the electronic potential to be sufficiently accurate for such systems. If, on the other hand, the assumption of nicely localized nuclear densities breaks down, one needs to incorporate a correlation contribution into v en Hc arising from the electron-nuclear interaction. 2 In the formalism presented here, the Berry-phase effects are assumed to be representable by a scalar nuclear potential. In view of the connection of the Berry phase to a vector potential in the nuclear Schrödinger equation 60, a multicomponent currentdensity description appears more appropriate to treat these effects. Thus, in the limit of classical nuclei, the Hxc energy functionals reduce to the classical electrostatic Hartree interactions and correlation contributions vanish 34. The corresponding electronic potentials, following from Eqs. 92 and 93, then read v en Hc class, = W en R= 0,r, 94 v U Hxc class, = U ext R= 0,r. 95 The first quantity is identical to the classical Coulomb field of the nuclei, whereas the second one describes the influence of potentials applied externally to the system. Both quantities together represent the external potential in BO-based DFT, reflecting again its coincidence with the MCDFT in the limit of classical nuclei. From this perspective, one also might consider Eq. 88 as the natural extension of Eq. 94 to